We present the design, implementation, and application of several families of fast multiplierless approximations of the discrete cosine transform (DCT) with the lifting scheme called the binDCT. These binDCT families are derived from Chen's (1977) and Loeffler's (1989) plane rotation-based factorizations of the DCT matrix, respectively, and the design approach can also be applied to a DCT of arbitrary size. Two design approaches are presented. In the first method, an optimization program is defined, and the multiplierless transform is obtained by approximating its solution with dyadic values. In the second method, a general lifting-based scaled DCT structure is obtained, and the analytical values of all lifting parameters are derived, enabling dyadic approximations with different accuracies. Therefore, the binDCT can be tuned to cover the gap between the Walsh-Hadamard transform and the DCT. The corresponding two-dimensional (2-D) binDCT allows a 16-bit implementation, enables lossless compression, and maintains satisfactory compatibility with the floating-point DCT. The performance of the binDCT in JPEG, H.263+, and lossless compression is also demonstrated.

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Fast multiplierless approximations of the DCT with the lifting scheme

Fixed-point optimization utility software is developed that can aid scaling and wordlength determination of digital signal processing algorithms written in C or C++. This utility consists of two programs: the range estimator and the fixed-point simulator. The former estimates the ranges of floating-point variables for purposes of automatic scaling, and the latter translates floating-point programs into fixed-point equivalents to evaluate the fixed-point performance by simulation. By exploiting the operator overloading characteristics of C++, the range estimation and the fixed-point simulation can be conducted by simply modifying the variable declaration of the original program. This utility is easily applicable to nearly all types of digital signal processing programs including nonlinear, time-varying, multirate, and multidimensional signal processing algorithms. In addition, this software can be used to compare the fixed-point characteristics of different implementation architectures. An optimization example for an 8/spl times/8 inverse discrete cosine transform (IDCT) architecture that conforms to the IEEE standard specifications is presented. The optimized results require 8% fewer gates when compared with the previous best implementation.

Fixed-point optimization utility for C and C++ based digital signal processing programs

An efficient method for computing the discrete cosine transform (DCT) is proposed. Based on direct decomposition of the DCT, the recursive properties of the DCT for an even length input sequence is derived, which is a generalization of the radix 2 DCT algorithm. Based on the recursive property, a new DCT algorithm for an even length sequence is obtained. The proposed algorithm is very structural and requires fewer computations when compared with others. The regular structure of the proposed algorithm is suitable for fast parallel algorithm and VLSI implementation.

Fast algorithm for computing discrete cosine transform

This chapter provides an overview of this book. The book presents the complete set of discrete cosine transforms (DCTs) and discrete sine transforms (DSTs) constituting the entire class of discrete sinusoidal unitary transforms, including their definitions, general mathematical properties, relations to the Karhunen-Loeve transform (KLT), with the emphasis on fast algorithms and integer approximations for their efficient implementations in the integer domain. The book covers various latest developments in DCTs and DSTs in a unified way, and it is essentially a detailed excursion on orthogonal/orthonormal DCT and DST matrices, their matrix factorizations, and integer approximations. For the DCT and DST to be viable, feasible, and practical, the fast algorithms are essential for their efficient implementation in terms of reduced memory, implementation complexity, and recursivity. Extensive definitions, principles, properties, signal flow graphs, derivations, proofs, and examples are provided throughout the book for proper understanding of the strengths and shortcomings of the spectrum of cosine/sine transforms and their application in diverse disciplines.

CHAPTER 1 – Discrete Cosine and Sine Transforms

This paper proposes a new framework for digital image processing; it relies on inexact computing to address some of the challenges associated with the discrete cosine transform (DCT) compression. The proposed framework has three levels of processing; the first level uses approximate DCT for image compressing to eliminate all computational intensive floating-point multiplications and executing the DCT processing by integer additions and in some cases logical right/left shifts. The second level further reduces the amount of data (from the first level) that need to be processed by filtering those frequencies that cannot be detected by human senses. Finally, to reduce power consumption and delay, the third level introduces circuit level inexact adders to compute the DCT. For assessment, a set of standardized images are compressed using the proposed three-level framework. Different figures of merits (such as energy consumption, delay, power-signal-to-noise-ratio, average-difference, and absolute-maximum-difference) are compared to existing compression methods; an error analysis is also pursued confirming the simulation results. Results show very good improvements in reduction for energy and delay, while maintaining acceptable accuracy levels for image processing applications.

Approximate DCT Image Compression Using Inexact Computing

A fixed-point-error analysis for several 1D fast DCT algorithms is presented. For comparison, a direct-form approach is also included in the investigation. A statistical model is used as the basis for predicting the fixed-point error in implementing the algorithms, and a suitable scaling scheme is selected to avoid overflow. Closed-form expressions for both the mean and variance for fixed-point error are derived and compared with experimental results. Simulation results show close agreement between theory and experiment, validating the analysis. The results show that one of the algorithms is better than others in terms of average SNR performance. Based on the 1D analysis, attempts are made to investigate the fixed-point-error analysis of the two-column approach for 2D DCT. It is found that the fixed-point-error characteristics of the row-column approach for 2D DCTs is very similar to that of their 1D counterparts. >

Il Dong Tun

Papers

On the fixed-point-error analysis of several fast DCT algorithms